Temporal Fusion Transformers for interpretable multi-horizon time series forecasting

Multi-horizon forecasting often contains a complex mix of inputs – including static (i.e. time-invariant) covariates, known future inputs, and other exogenous time series that are only observed in the past – without any prior information on how they interact with the target. Several deep learning methods have been proposed, but they are typically ‘black-box’ models that do not shed light on how they use the full range of inputs present in practical scenarios. In this paper, we introduce the Temporal Fusion Transformer (TFT) – a novel attention-based architecture that combines high-performance multi-horizon forecasting with interpretable insights into temporal dynamics. To learn temporal relationships at different scales, TFT uses recurrent layers for local processing and interpretable self-attention layers for long-term dependencies. TFT utilizes specialized components to select relevant features and a series of gating layers to suppress unnecessary components, enabling high performance in a wide range of scenarios. On a variety of real-world datasets, we demonstrate significant performance improvements over existing benchmarks, and highlight three practical interpretability use cases of TFT.     

Online distributed learning in wind power forecasting

Forecasting wind power generation up to a few hours ahead is of the utmost importance for the efficient operation of power systems and for participation in electricity markets. Recent statistical learning approaches exploit spatiotemporal dependence patterns among neighbouring sites, but their requirement of sharing confidential data with third parties may limit their use in practice. This explains the recent interest in distributed, privacy preserving algorithms for high-dimensional statistical learning, e.g. with auto-regressive models. The few approaches that have been proposed are based on batch learning. However, these approaches are potentially computationally expensive and do not allow for the accommodation of nonstationary characteristics of stochastic processes like wind power generation. This paper closes the gap between online and distributed optimisation by presenting two novel approaches that recursively update model parameters while limiting information exchange between wind farm operators and other potential data providers. A simulation study compared the convergence and tracking ability of both approaches. In addition, a case study using a large dataset from 311 wind farms in Denmark confirmed that online distributed approaches generally outperform existing batch approaches while preserving privacy such that agents do not have to actively share their private data.     

Machine learning methods for GEFCom2017 probabilistic load forecasting

This paper describes the preprocessing and forecasting methods used by team Orbuculum during the qualifying match of the Global Energy Forecasting Competition 2017 (GEFCom2017). Tree-based algorithms (gradient boosting and quantile random forest) and neural networks made up an ensemble. The ensemble prediction quantiles were obtained by a simple averaging of the ensemble members’ prediction quantiles. The result shows a robust performance according to the pinball loss metric, with the ensemble model achieving third place in the qualifying match of the competition.     

The theta model: a decomposition approach to forecasting

This paper presents a new univariate forecasting method. The method is based on the concept of modifying the local curvature of the time-series through a coefficient ‘Theta’ (the Greek letter θ), that is applied directly to the second differences of the data. The resulting series that are created maintain the mean and the slope of the original data but not their curvatures. These new time series are named Theta-lines. Their primary qualitative characteristic is the improvement of the approximation of the long-term behavior of the data or the augmentation of the short-term features, depending on the value of the Theta coefficient. The proposed method decomposes the original time series into two or more different Theta-lines. These are extrapolated separately and the subsequent forecasts are combined. The simple combination of two Theta-lines, the Theta=0 (straight line) and Theta=2 (double local curves) was adopted in order to produce forecasts for the 3003 series of the M3 competition. The method performed well, particularly for monthly series and for microeconomic data.     

DeepAR: Probabilistic forecasting with autoregressive recurrent networks

Probabilistic forecasting, i.e., estimating a time series’ future probability distribution given its past, is a key enabler for optimizing business processes. In retail businesses, for example, probabilistic demand forecasts are crucial for having the right inventory available at the right time and in the right place. This paper proposes DeepAR, a methodology for producing accurate probabilistic forecasts, based on training an autoregressive recurrent neural network model on a large number of related time series. We demonstrate how the application of deep learning techniques to forecasting can overcome many of the challenges that are faced by widely-used classical approaches to the problem. By means of extensive empirical evaluations on several real-world forecasting datasets, we show that our methodology produces more accurate forecasts than other state-of-the-art methods, while requiring minimal manual work.     

An empirical investigation of water consumption forecasting methods

Many regions on earth face daily limitations in the quantity and quality of the water resources available. As a result, it is necessary to implement reliable methodologies for water consumption forecasting that will enable the better management and planning of water resources. This research analyses, for the first time, a large database containing data from 2 million water meters in 274 unique postal codes, in one of the most densely populated areas of Europe, which faces issues of droughts and overconsumption in the hot summer months. Using the R programming language, we built and tested three alternative forecasting methodologies, employing univariate forecasting techniques including a machine-learning algorithm, with very promising results.     

Deep learning models for visibility forecasting using climatological data

Low visibility conditions affect safety and traffic operations, leading to adverse scenarios that often result in serious accidents. Due to the complexity and variability associated with modeling weather variables, visibility forecasting remains a highly challenging task and a matter of significant interest for transportation agencies nationwide. Given that the literature on single-step visibility forecasting is very scarce, this study explores the use of deep learning models for single-step visibility forecasting using time series climatological data. Five different deep learning models were developed, trained, and tested using data from two weather stations located in the US state of Florida, which is one of the top states nationwide dealing with low visibility problems. The authors provide discussions of the models’ results and areas for future research.     

Spatio-temporal probabilistic forecasting of wind power for multiple farms: A copula-based hybrid model

Accurate probabilistic forecasting of wind power output is critical to maximizing network integration of this clean energy source. There is a large literature on temporal modeling of wind power forecasting, but considerably less work combining spatial dependence into the forecasting framework. Through the careful consideration of the temporal modeling component, complemented by support vector regression of the temporal model residuals, this work demonstrates that a DVINE copula model most accurately represents the residual spatial dependence. Additionally, this work proposes a complete set of validation mechanisms for multi-h-step forecasts that, when considered together, comprehensively evaluate accuracy. The model and validation mechanisms are demonstrated in two case studies, totaling ten wind farms in the Texas electric grid. The proposed method outperforms baseline and competitive models, with an average Continuous Ranked Probability Score of less than 0.15 for individual farms, and an average Energy Score of less than 0.35 for multiple farms, over the 24-hour-ahead horizon. Results show the model’s ability to replicate the power output dynamics through calibrated and sharp predictive densities.     

Online hierarchical forecasting for power consumption data

This paper proposes a three-step approach to forecasting time series of electricity consumption at different levels of household aggregation. These series are linked by hierarchical constraints—global consumption is the sum of regional consumption, for example. First, benchmark forecasts are generated for all series using generalized additive models. Second, for each series, the aggregation algorithm ML-Poly, introduced by Gaillard, Stoltz, and van Erven in 2014, finds an optimal linear combination of the benchmarks. Finally, the forecasts are projected onto a coherent subspace to ensure that the final forecasts satisfy the hierarchical constraints. By minimizing a regret criterion, we show that the aggregation and projection steps improve the root mean square error of the forecasts. Our approach is tested on household electricity consumption data; experimental results suggest that successive aggregation and projection steps improve the benchmark forecasts at different levels of household aggregation.     

Kaggle forecasting competitions: An overlooked learning opportunity

We review the results of six forecasting competitions based on the online data science platform Kaggle, which have been largely overlooked by the forecasting community. In contrast to the M competitions, the competitions reviewed in this study feature daily and weekly time series with exogenous variables, business hierarchy information, or both. Furthermore, the Kaggle data sets all exhibit higher entropy than the M3 and M4 competitions, and they are intermittent.In this review, we confirm the conclusion of the M4 competition that ensemble models using cross-learning tend to outperform local time series models and that gradient boosted decision trees and neural networks are strong forecast methods. Moreover, we present insights regarding the use of external information and validation strategies, and discuss the impacts of data characteristics on the choice of statistics or machine learning methods. Based on these insights, we construct nine ex-ante hypotheses for the outcome of the M5 competition to allow empirical validation of our findings.     

Mixture of MLP-experts for trend forecasting of time series: A case study of the Tehran stock exchange

A new method for forecasting the trend of time series, based on mixture of MLP experts, is presented. In this paper, three neural network combining methods and an Adaptive Network-Based Fuzzy Inference System (ANFIS) are applied to trend forecasting in the Tehran stock exchange. There are two experiments in this study. In experiment I, the time series data are the Kharg petrochemical company’s daily closing prices on the Tehran stock exchange. In this case study, which considers different schemes for forecasting the trend of the time series, the recognition rates are 75.97%, 77.13% and 81.64% for stacked generalization, modified stacked generalization and ANFIS, respectively. Using the mixture of MLP experts (ME) scheme, the recognition rate is strongly increased to 86.35%. A gain and loss analysis is also used, showing the relative forecasting success of the ME method with and without rejection criteria, compared to a simple buy and hold approach. In experiment II, the time series data are the daily closing prices of 37 companies on the Tehran stock exchange. This experiment is conducted to verify the results of experiment I and to show the efficiency of the ME method compared to stacked generalization, modified stacked generalization and ANFIS.     

Neural basis expansion analysis with exogenous variables: Forecasting electricity prices with NBEATSx

We extend neural basis expansion analysis (NBEATS) to incorporate exogenous factors. The resulting method, called NBEATSx, improves on a well-performing deep learning model, extending its capabilities by including exogenous variables and allowing it to integrate multiple sources of useful information. To showcase the utility of the NBEATSx model, we conduct a comprehensive study of its application to electricity price forecasting tasks across a broad range of years and markets. We observe state-of-the-art performance, significantly improving the forecast accuracy by nearly 20% over the original NBEATS model, and by up to 5% over other well-established statistical and machine learning methods specialized for these tasks. Additionally, the proposed neural network has an interpretable configuration that can structurally decompose time series, visualizing the relative impact of trend and seasonal components and revealing the modeled processes’ interactions with exogenous factors. To assist related work, we made the code available in a dedicated repository.     

Short-term forecasting of crime

The major question investigated is whether it is possible to accurately forecast selected crimes 1 month ahead in small areas, such as police precincts. In a case study of Pittsburgh, PA, we contrast the forecast accuracy of univariate time series models with naïve methods commonly used by police. A major result, expected for the small-scale data of this problem, is that average crime count by precinct is the major determinant of forecast accuracy. A fixed-effects regression model of absolute percent forecast error shows that such counts need to be on the order of 30 or more to achieve accuracy of 20% absolute forecast error or less. A second major result is that practically any model-based forecasting approach is vastly more accurate than current police practices. Holt exponential smoothing with monthly seasonality estimated using city-wide data is the most accurate forecast model for precinct-level crime series.     

Online adaptive lasso estimation in vector autoregressive models for high dimensional wind power forecasting

Wind power forecasts with lead times of up to a few hours are essential to the optimal and economical operation of power systems and markets. Vector autoregression (VAR) is a framework that has been shown to be well suited to predicting for several wind farms simultaneously by considering the spatio-temporal dependencies in their time series. Lasso penalisation yields sparse models and can avoid overfitting the large numbers of coefficients in higher dimensional settings. However, estimation in VAR models usually does not account for changes in the spatio-temporal wind power dynamics that are related to factors such as seasons or wind farm setup changes, for example. This paper tackles this problem by proposing a time-adaptive lasso estimator and an efficient coordinate descent algorithm for updating the VAR model parameters recursively online. The approach shows good abilities to track changes in the multivariate time series dynamics on simulated data. Furthermore, in two case studies it shows clearly better predictive performances than the non-adaptive lasso VAR and univariate autoregression.     

A feature engineering approach to wind power forecasting: GEFCom 2012

This paper provides detailed information about team Leustagos’ approach to the wind power forecasting track of GEFCom 2012. The task was to predict the hourly power generation at seven wind farms, 48 hours ahead. The problem was addressed by extracting time- and weather-related features, which were used to build gradient-boosted decision trees and linear regression models. This approach achieved first place in both the public and private leaderboards.     

Nonlinear deterministic forecasting of daily dollar exchange rates

We perform out-of-sample predictions on several dollar exchange rate returns by using time-delay embedding techniques and a local linear predictor. We compared our predictions with those by a mean value predictor. Some of our predictions of the exchange rate returns outperform the predictions of the same series by the mean value predictor. However, these improvements were not statistically significant. Another interesting result in this paper which was obtained by using a recently developed technique of nonlinear dynamics is that all exchange rate return series we tested have a very high embedding dimension. Additionally, evidence indicates that these series are likely generated by high dimensional systems with measurement noise or by high dimensional nonlinear stochastic systems, that is, nonlinear deterministic systems with dynamic noise.     

Parameter-efficient deep probabilistic forecasting

Probabilistic time series forecasting is crucial in many application domains, such as retail, ecommerce, finance, and biology. With the increasing availability of large volumes of data, a number of neural architectures have been proposed for this problem. In particular, Transformer-based methods achieve state-of-the-art performance on real-world benchmarks. However, these methods require a large number of parameters to be learned, which imposes high memory requirements on the computational resources for training such models. To address this problem, we introduce a novel bidirectional temporal convolutional network that requires an order of magnitude fewer parameters than a common Transformer-based approach. Our model combines two temporal convolutional networks: the first network encodes future covariates of the time series, whereas the second network encodes past observations and covariates. We jointly estimate the parameters of an output distribution via these two networks. Experiments on four real-world datasets show that our method performs on par with four state-of-the-art probabilistic forecasting methods, including a Transformer-based approach and WaveNet, on two point metrics (sMAPE and NRMSE) as well as on a set of range metrics (quantile loss percentiles) in the majority of cases. We also demonstrate that our method requires significantly fewer parameters than Transformer-based methods, which means that the model can be trained faster with significantly lower memory requirements, which as a consequence reduces the infrastructure cost for deploying these models.     

A solution for M5 Forecasting - Uncertainty: Hybrid gradient boosting and autoregressive recurrent neural network for quantile estimation

This paper presents our 13th place solution to the M5 Forecasting - Uncertainty challenge and compares it against GoodsForecast’s second-place solution. This challenge aims to estimate the median and eight other quantiles of various product sales in Walmart. Both solutions handle the predictions of median and other quantiles separately. Our solution hybridizes LightGBM and DeepAR in various ways for median and quantile estimation, based on the aggregation levels of the sales. Similarly, GoodsForecast’s solution also utilized a hybrid approach, i.e., LightGBM for point estimation and a Histogram algorithm for quantile estimation. In this paper, the differences between the two solutions and their results are highlighted. Despite our solution only taking 13th place in the challenge with the competition metric, it achieves the lowest average rank based on the multiple comparisons with the best (MCB) test which implies the most accurate forecasts in the majority of the series. It also indicates better performance at the product-store aggregation level which comprises 30,490 (71.2% of all) series compared to most teams.     

A combination-based forecasting method for the M4-competition

Several researchers (Armstrong, 2001; Clemen, 1989; Makridakis and Winkler, 1983) have shown empirically that combination-based forecasting methods are very effective in real world settings. This paper discusses a combination-based forecasting approach that was used successfully in the M4 competition. The proposed approach was evaluated on a set of 100K time series across multiple domain areas with varied frequencies. The point forecasts submitted finished fourth based on the overall weighted average (OWA) error measure and second based on the symmetric mean absolute percent error (sMAPE).     

Forecasting ATM cash demands using a local learning model of cerebellar associative memory network

Forecasting cash demands at automatic teller machines (ATMs) is challenging, due to the heteroskedastic nature of such time series. Conventional global learning computational intelligence (CI) models, with their generalized learning behaviors, may not capture the complex dynamics and time-varying characteristics of such real-life time series data efficiently. In this paper, we propose to use a novel local learning model of the pseudo self-evolving cerebellar model articulation controller (PSECMAC) associative memory network to produce accurate forecasts of ATM cash demands. As a computational model of the human cerebellum, our model can incorporate local learning to effectively model the complex dynamics of heteroskedastic time series. We evaluated the forecasting performance of our PSECMAC model against the performances of current established CI and regression models using the NN5 competition dataset of 111 empirical daily ATM cash withdrawal series. The evaluation results show that the forecasting capability of our PSECMAC model exceeds that of the benchmark local and global-learning based models.